[GAP Forum] Help with recovering Lie brackets

[Alan Hylton]

Thank you! This is exactly what I needed!

Best,
agh

On Mon, Apr 26, 2021 at 3:20 PM Willem Adriaan De Graaf <
willem.degraaf at unitn.it> wrote:

> Dear Alan,
>
> You can use * to multiply elements of a Lie algebra. In the example you
> mention:
>
> gap> L:= SimpleLieAlgebra( "F", 4, Rationals );
> <Lie algebra of dimension 52 over Rationals>
> gap> c:= ChevalleyBasis(L);;
> gap> c[1][1]*c[1][2];
> 0*v.1
> gap> c[1][1]*c[1][3];
> (-1)*v.5
>
> StructureConstantsTable is applied to a basis of L:
>
> gap> T:= StructureConstantsTable( Basis(L) );;
> gap> T[1][3];
> [ [ 5 ], [ -1 ] ]
>
> The latter means that e1*e3 is equal to -e5 (where e1,e2,... denote the
> standard basis vectors).
>
> Best wishes,
>
> Willem
>
>
> On Sun, 25 Apr 2021 at 19:38, Alan Hylton <agh314 at lehigh.edu> wrote:
>
>> Hi everyone,
>>
>> I am brand new to GAP. I am trying to follow along the Lie algebra
>> documentation here:
>> https://www.gap-system.org/Manuals/doc/ref/chap64_mj.html
>>
>> Right now I can create my Lie algebra:
>>
>> gap> L := SimpleLieAlgebra( "F", 4, Rationals );
>>
>> and I can my positive/negative roots and Cartan:
>>
>> gap> ChevalleyBasis(L);
>> [ [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13,
>> v.14, v.15, v.16, v.17, v.18, v.19,
>>       v.20, v.21, v.22, v.23, v.24 ], [ v.25, v.26, v.27, v.28, v.29,
>> v.30,
>> v.31, v.32, v.33, v.34, v.35, v.36,
>>       v.37, v.38, v.39, v.40, v.41, v.42, v.43, v.44, v.45, v.46, v.47,
>> v.48 ], [ v.49, v.50, v.51, v.52 ] ]
>>
>> But I cannot figure out how to do basic computations - like [e1, e2].
>> There
>> is a function called LieBracket, and the documentation suggests that *
>> works, but I can't get either to work.
>>
>> I would also like to see if I can return a list of non-trivial bracket
>> relations, or structure constants. I found there is a function called
>> StructureConstantsTable, but it does not seem to work with L.
>>
>> I'm sorry to ask what is surely simple stuff - I just don't see it yet.
>>
>> Thanks!
>> agh
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>